In the above model, the evolution of the environment is subject to the influence of the players choosing to cooperate, similarly to previous works [1]}, [2]}. To derive the corresponding model via pairwise comparison dynamics, recall that the rate \(\phi _{ij}\)  is given by the proportional rule \(\phi _{ij} = [a_j - a_i]_+ = \sum _k [a_{jk} - a_{ik}]_+x_j\) . Therefore we have:
\(\begin{array}{lll}\epsilon \dot{x}_i = x_j [\phi _{ji}]_+ + (1 - x_i - x_j) [\phi _{3i}]_+ - x_i ( [\phi _{ij}]_+ + [\phi _{i3}]_+ ), \\\dot{n} = n (1 - n) [(1 + \lambda ) x_1 -1],\end{array}\) 
